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Non-Abelian Quantum Hall States and Fractional Statistics

Waheb Bishara

Published May 28, 2009• 1 citation
Publisher

Abstract

The discovery of the fractional quantum Hall effect stimulated the investigation of anyons, particles with fractional statistics which are neither bosons nor fermions. This thesis focuses on the study of quantum Hall states which may support non-Abelian anyons. We first address the validity of assumptions used in the numerical study of such states, and then proceed with analyzing different experiments which can detect non-Abelian fractional statistics. We quantitatively analyze the two-point contact interferometer experiment, which is hoped to display clear-cut, direct evidence of non-Abelian fractional statistics. We calculate the temperature and voltage dependence of the interference experiment outcome, and the signal attenuation due to finite temperature loss of coherence. We then analyze the edge theory of a family of non-Abelian quantum Hall states in the second Landau level, and examine the tunneling between these states and a quantum dot. This tunneling problem maps onto the multi-channel Kondo problem, and will allow distinguishing between different quantum Hall states. Finally, we use the same theoretical methods for analyzing Sagnac interference in the conductance of a carbon nanotube loop, a one-dimensional system.

DOI
10.7907/wqc4-w268

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Physical Review Letters•Oct 29, 2024•0 citations
The discovery of the fractional quantum Hall effect stimulated the investigation of anyons, particles with fractional statistics which are neither bosons nor fermions. This thesis focuses on the study of quantum Hall states which may support non-Abelian anyons. We first address the validity of assumptions used in the numerical study of such states, and then proceed with analyzing different experiments which can detect non-Abelian fractional statistics. We quantitatively analyze the two-point contact interferometer experiment, which is hoped to display clear-cut, direct evidence of non-Abelian fractional statistics. We calculate the temperature and voltage dependence of the interference experiment outcome, and the signal attenuation due to finite temperature loss of coherence. We then analyze the edge theory of a family of non-Abelian quantum Hall states in the second Landau level, and examine the tunneling between these states and a quantum dot. This tunneling problem maps onto the multi-channel Kondo problem, and will allow distinguishing between different quantum Hall states. Finally, we use the same theoretical methods for analyzing Sagnac interference in the conductance of a carbon nanotube loop, a one-dimensional system.
This thesis is primarily a study of the measurement theory of non-Abelian anyons through interference experiments. We give an introduction to the theory of anyon models, providing all the formalism necessary to apply standard quantum measurement theory to such systems. This formalism is then applied to give a detailed analysis of a Mach-Zehnder interferometer for arbitrary anyon models. In this treatment, we find that the collapse behavior exhibited by a target anyon in a superposition of states is determined by the monodromy of the probe anyons with the target. Such measurements may also be used to gain knowledge that would help to properly identify the anyon model describing an unknown system. The techniques used and results obtained from this model interferometer have general applicability, and we use them to also describe the interferometry measurements in a two point-contact interferometer proposed for non-Abelian fractional quantum Hall states. Additionally, we give the complete description of a number of important examples of anyon models, as well as their corresponding quantities that are relevant for interferometry. Finally, we give a partial classification of anyon models with small numbers of particle types.
We propose a method to extract the mutual exchange statistics of the anyonic excitations of a general Abelian fractional quantum Hall state, by comparing the tunneling characteristics of a quantum point contact in two different experimental conditions. In the first, the tunneling current between two edges at different chemical potentials is measured. In the second, one of these edges is strongly diluted by an earlier point contact. We describe the case of the dilute beam in terms of a time-domain interferometer between the anyons flowing along the edge and quasiparticle-quasihole excitations created at the tunneling quantum point contact. In both cases, temperature is kept large, such that the measured current is given to linear response. Remarkably, our proposal does not require the measurement of current correlations, and allows us to carefully separate effects of the fractional charge and statistics from effects of intra- and interedge interactions.
Non-Abelian anyons are fractional excitations of gapped topological models believed to describe certain topological superconductors or quantum Hall states. Here, we provide the first numerical evidence that they emerge as independent entities also in gapless electronic models. Starting from a multi-impurity multichannel chiral Kondo model, we introduce a novel mapping to a single-impurity model, amenable to Wilson's numerical renormalization group. We extract its spectral degeneracy structure and fractional entropy, and calculate the F matrices, which encode the topological information regarding braiding of anyons, directly from impurity spin-spin correlations. Impressive recent advances on realizing multichannel Kondo systems with chiral edges may thus bring anyons into reality sooner than expected.
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θθθ. θ=0,πθ= 0, πθ=0,π correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, θθθ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve non-abelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of non-abelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern--Simons terms. The crossings of one-dimensional anyons on a ring are uni-directional, such that a fractional phase θθθ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently the anyon behavior predicted for quasiparticles in the ν=1/3ν= 1/3ν=1/3 fractional quantum Hall state has been observed both in scattering and in interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
Abstract A collider where particles are injected onto a beam splitter from opposite sides has been used for identifying quantum statistics of identical particles. The collision leads to bunching of the particles for bosons and antibunching for fermions. In recent experiments, a collider was applied to a fractional quantum Hall regime hosting Abelian anyons. The observed negative cross-correlation of electrical currents cannot be understood with fermionic antibunching. Here we predict, based on a conformal field theory and a non-perturbative treatment of non-equilibrium anyon injection, that the collider provides a tool for observation of the braiding statistics of various Abelian and non-Abelian anyons. Its dominant process is not direct collision between injected anyons, contrary to common expectation, but braiding between injected anyons and an anyon excited at the collider. The dependence of the resulting negative cross-correlation on the injection currents distinguishes non-Abelian SU(2)k anyons, Ising anyons, and Abelian Laughlin anyons.
On the edge of certain fractional quantum Hall states, e.g., at 2/3 and 5/2 filling, a local fractional excitation, occurring by anyon tunneling at a quantum point contact, is further fractionalized into counterpropagating charge and neutral (Abelian or non-Abelian) anyonic excitations. We propose a scheme to detect the braiding statistics of the charge and neutral anyons separately. It is the injection of a dilute beam of a target (charge or neutral) anyon to a Fabry-Perot interferometer. The monodromy of the target anyon is obtained by comparing the amplitude and phase of the interference current with a reference signal of the same setup but without the injection. Our proposal relies on braiding between anyons on the edge, and applies even in the presence of bulk-edge couplings.